Unable to split an edge as expected
I hope I understood your question correctly, since the topic asks something else than the text. If I got it right, you want to know why you can not split a single selected edge.

I think the problem is that Blender can not connect two vertices with an edge twice.

If you only select one edge and try to split it, Blender would have to insert a second edge between its two vertices. My guess is that this collides with the internal data structure of Blender.

To solve this problem you would have to insert another vertex along this edge first, so that after splitting you will get a "ring" of four edges. Then there are no two vertices connect twice and Blender is happy.